Math Help

Archive for the ‘Integration’ Category

Question: How to find the Integral of csc x?

How do you get the integral of csc x? I needed this for a problem, I tried by parts and got a different answer from the book (the given answer is ln (sinx / (cosx + 1). It should be possible to do using only basic methods of Integration, as it was in the problems sections of the chapter before integral by parts. All help appreciated.

Answer: integral csc x

mutiply by , -cscx-cotx
so
-integral csc x*(-cscx-cotx)/(cscx+cotx)
let u= cscx +cotx,
du=-cscxcotx-csc^2x
integral – du/u
-ln u
-ln cscx +cotx

Vector Integral Calculus – How to find Lines of Force p1

Question: a confusion in Integration maths?

integrate (x^2+1)^5

(maybe writing it down will help

Answer: /
either expand or use trig sub
x=tany
dx=sec^2ydy
sub
intsec^7ydy
use
sec^n(y)dy=1/(n-1)*sec^(n-2)xtanx+
(n-2)/(n-1)*intsec^(n-2)xdx

Integration by Parts

Question: Help With Math integral story problem?

The book states:

It is estimated that T years from now the value of a certain parcel of land will be increasing at the rate of V’(T) dollars per year. Find an expression for the amount by which the value of the land will increase during the next 5 years.

If you could show me how you got to your answer, that would be great. Thanks!

Answer: Since V ‘(t) is the rate of increase, the actual amount of increase will be an integral of V ‘ (t). You can derive the answer by appealing to Riemann sums. Not sure what this particular author is after. If today corresponds to T=0, then the amount of increase during the next five years would be

integral(T=0 to 5) V ‘ (T) dT = V(5) – V(0)

where the last term on the right is due to the fundamental theorem of calculus.

To derive this result from “first principles” (if that is a good way to put it), we can partition the next five years—the interval from T=0 to T=5—into n small time increments of length δT (delta T). This partition would give us the times

T0=0,
T1=0 + δT = δT,
T2 = T1+δT = 2δT
…
Tn = n δT.

On each little segment, we could estimate that the rate is constant, say V ‘(T) = V ‘ (Ti). The amount of increase over each time interval would be the product of the rate times the amount of time

V ‘ (Ti) δT.

The total amount of increase over 5 years is the sum of all of these

Σ V ‘ (Ti) δT summing over i = 1 to n.

Take the limit as δT becomes infinitesimal, and you get the integral above.

Calculus Help: Integrals 1